Question

A set of data values is normally distributed with a mean of 90 and a standard deviation of 4. Give the standard score and approximate percentile for a data value of 89.

Answer 1

Answer:

Standard score of -0.25.

A data value of 89 is approximately in the 40th percentile.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 90, \sigma = 4$

Give the standard score and approximate percentile for a data value of 89.

Z when X = 89

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{89 - 90}{4}$

$Z = -0.25$

$Z = -0.25$ has a pvalue of 0.4013.

So a data value of 89 is approximately in the 40th percentile.